Now that we have a flavour for the variety and severity of the
selection effects that plague the observed sample of pulsars, how
do we decouple these effects to form a less biased picture of the
true population of objects? A very useful technique, first
employed by Phinney & Blandford and Vivekanand &
Narayan [191,
259], is to define a scaling factor
as the ratio of the total Galactic volume weighted by pulsar
density to the volume in which a pulsar could be detected by the
surveys:
In this expression,
is the assumed pulsar distribution in terms of galactocentric
radius
R
and height above the Galactic plane
z
. Note that
is primarily a function of period
P
and luminosity
L
such that short period/lowluminosity pulsars have smaller
detectable volumes and therefore higher
values than their long period/highluminosity counterparts. This
approach is similar to the classic
technique first used to correct observationallybiased samples
of quasars [211].
This technique can be used to estimate the total number of
active pulsars in the Galaxy. In practice, this is achieved by
calculating
for each pulsar separately using a Monte Carlo simulation to
model the volume of the Galaxy probed by the major surveys [170]. For a sample of
observed pulsars above a minimum luminosity
, the total number of pulsars in the Galaxy with luminosities
above this value is simply
where
f
is the modeldependent ``beaming fraction'' discussed below in
§
3.2.3
. Monte Carlo simulations of the pulsar population incorporating
the aforementioned selection effects have shown this method to be
reliable, as long as
is reasonably large [131].
For small samples of observationallyselected objects, the
detected sources are likely to be those with largerthanaverage
luminosities. The sum of the scale factors (Equation (5)), therefore, will tend to underestimate the true size of the
population. This ``smallnumber bias'' was first pointed out by
Kalogera et al. [112,
113] for the sample of double neutron star binaries where we know of
only three clearcut examples (§
3.4.1). Only when the number of sources in the sample gets past 10 or
so does the sum of the scale factors become a good indicator of
the true population size.
Figure 15:
Smallnumber bias of the scale factor estimates derived from a
synthetic population of sources where the true number of sources
is known. Left: An edgeon view of a model Galactic source
population. Right: The thick line shows
, the true number of objects in the model Galaxy, plotted against
, the number detected by a fluxlimited survey. The thin solid
line shows
, the median sum of the scale factors, as a function of
from a large number of MonteCarlo trials. Dashed lines show 25
and 75% percentiles of the
distribution.
The ``beaming fraction''
f
in Equation (5) is simply the fraction of
steradians swept out by the radio beam during one rotation. Thus
f
gives the probability that the beam cuts the lineofsight of an
arbitrarily positioned observer. A naïve estimate of
f
is 20%; this assumes a beam width of
and a randomly distributed inclination angle between the spin
and magnetic axes [238]. Observational evidence suggests that shorter period pulsars
have wider beams and therefore larger beaming fractions than
their longperiod counterparts [171,
149,
32,
231]. It must be said, however, that a consensus on the beaming
fractionperiod relation has yet to be reached. This is shown in
Fig.
16
where we compare the period dependence of
f
as given by a number of models. Adopting the Lyne &
Manchester model, pulsars with periods
ms beam to about 30% of the sky compared to the Narayan &
Vivekanand model in which pulsars with periods below 100 ms
beam to the entire sky.
Figure 16:
Beaming fraction plotted against pulse period for four
different beaming models: Tauris & Manchester 1998 (TM88; [231]), Lyne & Manchester 1988 (LM88; [149]), Biggs 1990 (JDB90; [32]) and Narayan & Vivekanand 1983 (NV83; [171]).
When most of these beaming models were originally proposed,
the sample of millisecond pulsars was
5 and hence their predictions about the beaming fractions of
shortperiod pulsars relied largely on extrapolations from the
normal pulsars. A recent analysis of a large sample of
millisecond pulsar profiles by Kramer et al. [122] suggests that the beaming fraction of millisecond pulsars lies
between 50 and 100%.

Binary and Millisecond Pulsars at the New Millennium
Duncan R. Lorimer
http://www.livingreviews.org/lrr20015
© MaxPlanckGesellschaft. ISSN 14338351
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livrev@aeipotsdam.mpg.de
